On real-valued SDE and nonnegative-valued SDE population models with demographic variability

Area of Science:

• Mathematical Biology
• Stochastic Processes
• Population Dynamics

Background:

• Population dynamics are often modeled using continuous-time Markov chain (CTMC) models with discrete random variables.
• Approximating CTMC models with continuous variables yields Itô stochastic differential equations (SDEs).
• Existing SDE models may produce infeasible non-real or negative population solutions.

Purpose of the Study:

• To derive novel SDE systems that guarantee real-valued and nonnegative solutions for population dynamics.
• To address the limitations of existing SDE models in population modeling.

Main Methods:

• Developed new SDE systems by assuming nonnegative reaction rates, assigning probability zero to negative rates.
• Modified diffusion coefficients near zero population sizes to ensure nonnegative solutions.
• Applied these methods to several population dynamic problems for validation.

Main Results:

• Successfully derived SDE models with guaranteed real-valued solutions.
• Ensured nonnegative solutions by adjusting diffusion coefficients for small population sizes.
• Demonstrated the feasibility and biological realism of the new SDE models.

Conclusions:

• The developed SDEs provide a more robust framework for modeling population dynamics with demographic variability.
• The methodology ensures biologically realistic, nonnegative population sizes, overcoming limitations of previous models.
• These SDEs offer a reliable tool for analyzing population dynamics where solutions must remain real and nonnegative.